## Profile

Qiaochu Yuan
Worked at Machine Intelligence Research Institute
Attends University of California, Berkeley
Lives in Berkeley, CA
2,313 followers|159,467 views

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### Qiaochu Yuan

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Let $latex k$ be a commutative ring and let $latex A$ be a $latex k$-algebra. In this post we’ll investigate a condition on $latex A$ which generalizes the condition that $latex A$ is a finit…
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### Qiaochu Yuan

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Mathematicians are very fond of thinking about algebras. In particular, it’s common to think of commutative algebras as consisting of functions of some sort on spaces of some sort. Less commo…
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### Qiaochu Yuan

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### Qiaochu Yuan

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Here's a dumb riddle. We all know that a meromorphic function on a Riemann surface is a function to CP^1. We also all know that meromorphic functions on a Riemann surface form a field, and in particular a ring. So... why isn't CP^1 a ring object in the category of Riemann surfaces?

(It took me an embarrassingly long time to figure this out.)﻿
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Is it because the point is not a Riemann surface?﻿

### Qiaochu Yuan

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Once upon a time I imagine people were very happy to think of Lie algebras as "infinitesimal groups," but presumably when infinitesimals fell out of favor this interpretation did too. In this post ...
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You can write code to "implement" a Lie algebra by writing polymorphic code for the group and replacing the base field of reals with the reals extended by infintesimals: http://blog.sigfpe.com/2008/04/infinitesimal-rotations-and-lie.html

The variable I call 'e' gives the comultiplication.﻿

### Qiaochu Yuan

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Previously we claimed that if you want to check whether a category $latex C$ "behaves like a category of spaces," you can try checking whether it's distributive. The goal of today's post is to just...
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Have them in circles
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### Qiaochu Yuan

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Previously we suggested that if we think of commutative algebras as secretly being functions on some sort of spaces, we should correspondingly think of cocommutative coalgebras as secretly being di…
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Here's a fun question: is the empty function (from the empty set to itself) constant? Is it locally constant? ﻿
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Presumably a constant function f:X->Y should factor through pt->Y uniquely. If that's the only definition, and X is empty, then "f is constant" iff |Y| = 1. Which is pretty weird!﻿

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Very nice visualization of prime factorizations. ﻿
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iirc, this is included in http://incrediblenumbersapp.com/, which is a nice app for kids﻿

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It's common to think of monads as generalized algebraic theories; the most familiar examples, such as the monads on $latex \text{Set}$ encoding groups, rings, and so forth, have this flavor. Howeve...
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### Qiaochu Yuan

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Ah, technology. It's a blessing and a curse. To help get more people to read your posts is pure pleasure on my side! In that reign, I tried some categorification-without-looking-at-literature myself, to come up with an alternative to burritos. And these notes want out, if I'd only dare to post them. Maybe more programmers people would like categorification if they knew more...

There. It's promised, nothing can go wrong now. And maybe it even works and someone shows up here for better, or for more.﻿

### Qiaochu Yuan

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Among all of the standard algebraic structures that a student typically encounters in an introduction to abstract algebra (groups, rings, fields, modules), commutative rings are somehow special: th...
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People
Have them in circles
2,313 people
Work
Occupation
Employment
• Machine Intelligence Research Institute
Visiting Fellow, 2013 - 2013
• PROMYS
Counselor, 2012 - 2012
• Stack Overflow, Inc.
Intern, 2011 - 2011
Places
Currently
Berkeley, CA
Previously
Boston, MA - New York, NY - Bellevue, WA - Nanjing, China - Singapore - Vancouver, WA - Cambridge, UK
Story
Introduction
I'm a third-year graduate student in mathematics at UC Berkeley.
Bragging rights
USA Mathematical Olympiad Honorable Mention (2006), Siemens-Westinghouse Science and Technology Competition Semifinalist (2007), Intel Science Talent Search Finalist (2008), William Lowell Putnam Mathematical Competition Honorable Mention (2009), NSF Fellow (2012)
Education
• University of California, Berkeley
Mathematics, 2012 - present
• Massachusetts Institute of Technology
Mathematics, 2008 - 2012
• University of Cambridge
Mathematics, 2010 - 2011